The Einstein-Podolsky-Rosen Argument in Quantum Theory

In the May 15, 1935 issue of Physical Review Albert
Einstein co-authored a paper with his two postdoctoral research
associates at the Institute for Advanced Study, Boris Podolsky and
Nathan Rosen. The article was entitled “Can Quantum Mechanical
Description of Physical Reality Be Considered Complete?” (Einstein
et al. 1935). Generally referred to as “EPR”,
this paper quickly became a centerpiece in the debate over the
interpretation of the quantum theory, a debate that continues
today. The paper features a striking case where two quantum systems
interact in such a way as to link both their spatial coordinates in a
certain direction and also their linear momenta (in the same
direction). As a result of this “entanglement”,
determining either position or momentum for one system would fix
(respectively) the position or the momentum of the other. EPR use this
case to argue that one cannot maintain both an intuitive condition of
local action and the completeness of the quantum description by means
of the wave function. This entry describes the argument of that 1935
paper, considers several different versions and reactions, and
explores the ongoing significance of the issues they raise.

By 1935 the conceptual understanding of the quantum theory was
dominated by Niels Bohr's ideas concerning complementarity. Those ideas
centered on observation and measurement in the quantum domain.
According to Bohr's views at that time, observing a quantum object
involves an uncontrollable physical interaction with a classical measuring
device that
affects both systems. The picture
here is of a tiny object banging into a big apparatus. The effect
this produces on the measuring instrument is what issues in the
measurement “result” which, because it is uncontrollable, can only be
predicted statistically. The effect experienced by the quantum
object restricts those quantities that can be co-measured with
precision. According to complementarity when we observe the position of
an object, we affect its momentum uncontrollably. Thus we cannot
determine precisely both position and momentum. A similar situation
arises for the simultaneous determination of energy and time. Thus
complementarity involves a doctrine of uncontrollable physical interaction
that, according to Bohr, underwrites the Heisenberg
uncertainty relations and is also the source of the statistical
character of the quantum theory. (See the entries on the
Copenhagen Interpretation
and the
Uncertainty Principle.)

Initially Einstein was enthusiastic about the quantum theory. By 1935,
however, his enthusiasm for the theory had given way to a sense of
disappointment. His reservations were twofold. Firstly, he felt the
theory had abdicated the historical task of natural science to provide
knowledge of significant aspects of nature that are independent of
observers or their observations. Instead the fundamental understanding
of the wave function (alternatively, the “state function”,
“state vector”, or “psi-function”) in quantum
theory was that it only treated the outcomes of measurements (via
probabilities given by the Born Rule). The theory was simply silent
about what, if anything, was likely to be true in the absence of
observation. That there could be laws, even probabilistic laws, for
finding things if one looks, but no laws of any sort for how things
are independently of whether one looks, marked quantum theory as
irrealist. Secondly, the quantum theory was essentially statistical.
The probabilities built into the state function were fundamental and,
unlike the situation in classical statistical mechanics, they were not
understood as arising from ignorance of fine details. In this sense
the theory was indeterministic. Thus Einstein began to probe how
strongly the quantum theory was tied to irrealism and
indeterminism.

He wondered whether it was possible, at least in principle, to ascribe
certain properties to a quantum system in the absence of
measurement. Can we suppose, for instance, that the decay of an atom
occurs at a definite moment in time even though such a definite decay
time is not implied by the quantum state function? That is, Einstein
began to ask whether the quantum mechanical description of reality was
complete. Since Bohr's complementarity provided strong support both
for irrealism and indeterminism and since it played such a dominant
role in shaping the prevailing attitude toward quantum theory,
complementarity became Einstein's first target.
In particular, Einstein had reservations about the
uncontrollable physical effects invoked by Bohr in the context of
measurement interactions, and about their role in fixing the
interpretation of the wave function. EPR was intended to support those
reservations in a particularly dramatic way.

Max Jammer (1974, pp. 166–181) describes the EPR paper as originating
with Einstein's reflections on a thought experiment he proposed during
discussions at the 1930 Solvay conference. The experiment imagines a
box that contains a clock set to time precisely the release (in the
box) of a photon with determinate energy. If this were feasible, it
would appear to challenge the unrestricted validity of the Heisenberg
uncertainty relation that sets a lower bound on the simultaneous
uncertainty of energy and time. (See the entry on
the Uncertainty Principle and also
Bohr 1949, who describes the discussions at the 1930 conference.) The
uncertainty relations, understood not just as a prohibition on what is
co-measurable, but on what is simultaneously real, were a central
component in the irrealist interpretation of the wave function. Jammer
(1974, p. 173) describes how Einstein's thinking about this
experiment, and Bohr's objections to it, evolved into a different
photon-in-a-box experiment, one that allows an observer to determine
either the momentum or the position of the photon indirectly, while
remaining outside, sitting on the box. Jammer associates this with the
distant determination of either momentum or position that, we shall
see, is at the heart of the EPR paper. Carsten Held (1998) cites a
related
correspondence with Paul Ehrenfest
from 1932 in which Einstein described an arrangement for the indirect
measurement of a particle of mass m using correlations with a
photon established through Compton scattering. Einstein's reflections
here foreshadow the argument of EPR, along with noting some of its
difficulties.

Thus without an experiment on m it is possible to
predict freely, at will, either the momentum or the
position of m with, in principle, arbitrary
precision. This is the reason why I feel compelled to ascribe
objective reality to both. I grant, however, that it is not
logically necessary. (Held 1998, p. 90)

Whatever their precursors, the ideas that found their way into EPR
were worked out in a series of meetings with Einstein and his two
assistants, Podolsky and Rosen. The actual text, however, was written
by Podolsky and, apparently, Einstein did not see the final draft
(certainly he did not correct it) before Podolsky submitted the paper
to Physical Review
in March of 1935. It was sent for publication the day after it
arrived. Upon seeing the published version, Einstein
complained that his central concerns were obscured by Podolsky's exposition.

For reasons of language this [paper] was written by
Podolsky after several discussions. Still, it did not come out as well
as I had originally wanted; rather, the essential thing was, so to
speak, smothered by the formalism [Gelehrsamkeit]. (Letter from
Einstein to Erwin Schrödinger, June 19, 1935. In Fine 1996, p.
35.)

Thus in discussing the argument of EPR we should consider both the
argument in Podolsky's text and lines of argument that Einstein himself
offers. We should also consider an argument presented in Bohr's reply to
EPR, which is possibly the best known version, although it differs
from the others in important ways.

The EPR text is concerned, in the first instance, with the logical
connections between two assertions. One asserts that quantum mechanics
is incomplete. The other asserts that incompatible quantities (those
whose operators do not commute, like a coordinate of position and
linear momentum in that direction) cannot have simultaneous “reality”
(i.e., simultaneously real values). The authors assert as a first premise,
later to be justified, that one or another of these must hold. It
follows that if quantum mechanics were complete (so that the first
option failed) then the second option would hold; i. e., incompatible
quantities
cannot have real values simultaneously. However they also
take as a second premise (also to be justified) that if quantum
mechanics were complete, then incompatible quantities (in particular
position and momentum) could indeed have simultaneous, real values.
They conclude that quantum mechanics is incomplete. The conclusion
certainly follows since otherwise (if the theory were complete)
one would have a contradiction. Nevertheless the argument is highly
abstract and formulaic and even at this point in its development one
can readily appreciate Einstein's disappointment with it.

EPR now proceed to establish the two premises, beginning with a
discussion of the idea of a complete theory. Here they offer only a
necessary condition; namely, that for a complete theory “every element
of the physical reality must have a counterpart in the physical
theory.” Although they do not specify just what an “element of physical
reality” is they use that expression when referring to the values of
physical quantities (positions, momenta, and so on) provided the following
sufficient condition holds (p. 777):

If, without in any way disturbing a system, we can predict
with certainty (i.e., with probability equal to unity) the value of a
physical quantity, then there exists an element of reality
corresponding to that quantity.

This sufficient condition for an "element of reality" is often referred to as the EPR
Criterion of Reality.

With these terms in place it is easy to show that if, say, the
values of position and momentum for a quantum system were
real simultaneously (i.e., were elements of reality) then the
description provided by the wave function of the system would be
incomplete, since no wave function contains counterparts for both
elements. (Technically, no state function—even an improper one,
like a delta function—is a simultaneous eigenstate for both
position and momentum.) Thus they establish the first premise: either
quantum theory is incomplete or there can be no simultaneously real
values for incompatible quantities. They now need to show that if
quantum mechanics were complete, then incompatible quantities could
have simultaneous real values, which is the second premise. This,
however, is not easily established. Indeed what EPR proceed to do is
odd. Instead of assuming completeness and on that basis deriving
that incompatible quantities can have real values simultaneously , they
simply set out to derive the latter assertion without any completeness
assumption at all. This “derivation” turns out to be the heart of the
paper and its most controversial part. It attempts to show that in
certain circumstances a quantum system can have simultaneous values for
incompatible quantities (once again, for position and momentum), where
these values also pass the Reality Criterion's test for being “elements
of reality”.

They proceed by sketching a thought experiment. In the experiment two
quantum systems interact in such a way that two conservation laws hold
following their interaction. One is the conservation of relative
position. If we imagine the systems located along the x-axis,
then if one of the systems (we can call it Albert's) were found at
position q along the axis at a certain time, the other system
(call it Niels') would be found then a fixed distance d away,
say at q′ = q − d, where
we may suppose that the distance
d between q and q′ is
substantial. The other conservation law is that the total linear
momentum (along that same axis) is always zero. So when the momentum
of Albert's system along the x-axis is determined to
be p, the momentum of Niels' system would be found to be
−p. The paper constructs an explicit wave function for
the combined (Albert+Niels) system that satisfies both conservation
principles. Although commentators later raised questions about the
legitimacy of this wave function, it does appear to satisfy the two
conservation principles at least for a moment (Jammer 1974,
pp. 225–38; see also Halvorson 2000). In any case, one can model the
same conceptual situation in other cases that are clearly well defined
quantum mechanically (see Section 3.1).

At this point of the argument (p. 779) EPR make two critical
assumptions, although they do not call special attention to them. (For
the significance of these assumptions in Einstein's thinking see
Howard 1985 and also section 5 of the entry on
Einstein.) The
first assumption (separability) is that at the time when
measurements will be performed on Albert's system there is some reality
that pertains to Niels' system alone. In effect, they assume that
Niels' system maintains its separate identity even though it is correlated
with Albert's. They need this assumption to make sense of another. The
second assumption is that of locality. This supposes that “no
real change can take place” in Niels' system as a consequence of a
measurement made on Albert's system. They gloss this by saying “at the
time of measurement the two systems no longer interact.” Notice that
this is not a general principle of no-disturbance, but rather a
principle only governing disturbance or change in what is real with respect to Niels'
system. On the basis of these two assumptions they conclude that Niels'
system can have real values (“elements of reality”) for both position
and momentum simultaneously. There is no detailed argument for this in
the text. Instead they use these two assumptions to show how one could
be led to assign both a position eigenstate and a momentum eigenstate
to Niels' system, from which the simultaneous attribution of elements
of reality is supposed to follow. Since this is the central and most
controversial part of the paper, it pays to go slowly here in trying to
reconstruct an argument on their behalf.

One attempt might go as follows. Separability holds that some
reality pertains to Niels' system. Suppose that we measure, say, the
position of Albert's system. The reduction of the state function for
the combined systems then yields a position eigenstate for Niels'
system. That eigenstate applies to the reality there and that
eigenstate enables us to predict a determinate position for Niels'
system with probability one. Since that prediction only depends on a
measurement made on Albert's system, locality implies that the
prediction of the position of Niels' system does not involve any change
in the reality of Niels' system. If we interpret this as meaning that
the prediction does not disturb Niels' system, all the pieces are in
place to apply
The Criterion of Reality.
It certifies
that the predicted position value, corresponding to the position
eigenstate, is an element of the reality that pertains to Niels'
system. One could argue similarly with respect to momentum.

This line of argument, however, is deceptive and contains a serious
confusion. It occurs right after we apply locality to conclude that the
measurement made on Albert's system does not affect the reality
pertaining to Niels' system. For, recall, we have not yet determined
whether the position inferred for Niels' system is indeed an “element”
of that reality. Hence it is still possible that the measurement of
Albert's system, while not disturbing the reality pertaining
to Niels' system, does disturb its position. To take the extreme case;
suppose, for example, that the measurement of Albert's system somehow
brings the position of Niels' system into being, or suddenly makes it
well defined, and also allows us to predict it with certainty. It would
then follow from locality that the position of Niels' system is
not an element of the reality of that system, since it can be
affected at a distance. But, reasoning exactly as above, the Criterion
would still hold that the position of Niels' system is an
element of the reality there, since it can be predicted with certainty
without disturbing the reality of the system. What has gone
wrong? It is that the Criterion provides a sufficient condition for
elements of reality and locality provides a necessary condition. But,
as above, there is no guarantee that these conditions will always match
consistently. To ensure consistency we need to be sure that what the
Criterion certifies as real is not something that can be influenced at
a distance. One way to do this, which seems to be implicit in the EPR
paper, would be to interpret locality in the EPR situation in such a
way that measurements made on one system are understood not to disturb
those quantities on the distant, unmeasured system whose values can be
inferred from the reduced state of that system. Given the two
conservation laws satisfied in the EPR situation, this extended way of
understanding locality allows the Criterion to certify that position,
as well as momentum, when inferred for Niels' system, are real
there.

As EPR point out, however, position and momentum cannot be measured
simultaneously. So even if each can be shown to be real in distinct
contexts of measurement, are both real at the same time? EPR answers
“yes”, but it does not provide a clear rationale for that
conclusion. Here's one suggestion. (Dickson 2004 analyzes some of the
modal principles involved and suggests another route, which he
criticizes. Hooker 1972 is a comprehensive discussion that identifies
several generically different ways to make the case.) Suppose the
logical force of locality is to decontextualize the reality of Niels'
system from goings on at Albert's. Clearly when we infer from a
certain measurement made on Albert's system that Niels' system has an
element of reality, locality kicks in and guarantees that Niels'
system would have had that same element of reality even in the absence
of the measurement on Albert's system. So suppose, then, the
circumstance where we do not make that measurement. Could
that absence of a measurement on Albert's system affect what is real
on Niels' system? The suggestion is that we allow locality to kick in
here as well, with the answer “no”. Put differently, we
suggest that locality entitles us to conclude that Niels' system has a
real position provided the conditional assertion “If a position
measurement is performed on Albert's system, then Niels' system has a
real position” holds. Similarly, Niels' system has a real
momentum provided the conditional “If a momentum measurement is
performed on Albert's system, then Niels' system has a real
momentum” holds. (This is exactly how Einstein 1948 argues. See
Born 1971, p. 172.) Of course these conclusions presuppose that there
are no interfering factors operating locally on Niels' system, such as
a competing measurement. As we have seen, given separability,
locality and the Criterion of Reality both conditionals hold. Hence,
in the absence of interference, locality implies that Niels' system
has real values of both position and momentum simultaneously, even
though no simultaneous measurement of position and momentum is
allowed. (Reciprocally, so would Albert's system, provided we made no
interfering measurements there.)

In the penultimate paragraph of EPR (p. 780) they address the
problem of getting real values for incompatible quantities
simultaneously.

Indeed one would not arrive at our conclusion if one
insisted that two or more physical quantities can be regarded as
simultaneous elements of reality only when they can be
simultaneously measured or predicted. … This makes the
reality [on the second system] depend upon the process of measurement
carried out on the first system, which does not in any way disturb the
second system. No reasonable definition of reality could be expected to
permit this.

The unreasonableness to which EPR allude in making “the reality [on
the second system] depend upon the process of measurement carried out
on the first system, which does not in any way disturb the second
system” is just the unreasonableness that would be involved in
renouncing locality understood as above. For it is locality that enables one
to
overcome the incompatibility of position and momentum measurements of
Albert's system by requiring their joint consequences for Niels' system
to be incorporated in a single, stable reality there. If we recall
Einstein's acknowledgment to Ehrenfest that
getting simultaneous position and momentum was “not logically
necessary”, we can see how EPR respond by making
it become necessary once locality is assumed.

Here, then, are the key features of EPR.

EPR is about the interpretation of state vectors (“wave
functions”) and employs the standard state vector reduction formalism
(von Neumann's “projection postulate”).

The Criterion of Reality
is only used to check,
after state vector reduction assigns an eigenstate to the
unmeasured system, that the associated eigenvalue constitutes an
element of reality.

(Separability)
EPR make the
tacit assumption that, when they are spatially separated, some
“reality” pertains to both components of the combined
system.

(Locality)
EPR assume a
principle of locality according to which, if two systems are far enough
apart, the measurement (or absence of measurement) of one system does not
directly affect the
reality that pertains to the unmeasured system. (This
non-disturbance is understood to include those quantities on the
distant, unmeasured system whose values can be inferred from the
reduced state of that system.)

Locality is critical in guaranteeing that simultaneous position and
momentum values can be assigned to the unmeasured system even though
position and momentum cannot be measured simultaneously on the other
system.

Assuming separability and locality, the demonstration of
simultaneous position and momentum values depends on the state vector
descriptions in conjunction with the Criterion of Reality.

In summary, the argument of EPR shows that if interacting systems
satisfy separability and locality, then the description of systems
provided by state vectors is not complete. This conclusion rests on a
common interpretive principle, state vector reduction, and on the
Criterion of Reality.

The EPR experiment with interacting systems accomplishes a form of
indirect measurement. The direct measurement of Albert's system yields
information about Niels' system; it tells us what we would find if we
were to measure there directly. But it does this at-a-distance, without
any further physical interaction taking place between the two systems.
Thus the thought experiment at the heart of EPR undercuts the picture
of measurement as necessarily involving a tiny object banging into a
large measuring instrument. If we look back at
Einstein's reservations
about complementarity, we can
appreciate that by focusing on a non-disturbing kind of measurement the
EPR argument targets Bohr's program for explaining central conceptual
features of the quantum theory. For that program relied on
uncontrollable interaction with a measuring device as a necessary feature of
any measurement
in the quantum domain. Nevertheless the cumbersome machinery employed
in the EPR paper makes it difficult to see what is central. It
distracts from rather than focuses on the issues. That was Einstein's
complaint about Podolsky's text in his June 19, 1935 letter to
Schrödinger. Schrödinger responded on July 13 reporting
reactions to EPR that vindicate Einstein's concerns. With reference to
EPR he wrote:

I am now having fun and taking your note to its source to
provoke the most diverse, clever people: London, Teller, Born, Pauli,
Szilard, Weyl. The best response so far is from Pauli who at least
admits that the use of the word “state” [“Zustand”] for the
psi-function is quite disreputable. What I have so far seen by way of
published reactions is less witty. … It is as if one person
said, “It is bitter cold in Chicago”; and another answered, “That is a
fallacy, it is very hot in Florida.” (Fine 1996, p. 74)

If the argument developed in EPR has its roots in the 1930 Solvay
conference, Einstein's own approach to issues at the heart of EPR has
a history that goes back to the 1927 Solvay conference. (Bacciagaluppi
and Valentini 2009, pp. 198–202, would even trace it back to 1909 and
the localization of light quanta.) At that 1927 conference Einstein
made a short presentation during the general discussion session, where
he focused on problems of interpretation associated with the collapse
of the wave function. He imagines a situation where electrons pass
through a small hole and are dispersed uniformly in the direction of a
screen of photographic film shaped into a large hemisphere that
surrounds the hole. On the supposition that quantum theory offers a
complete account of individual processes then, in the case of
localization, why does the whole wave front collapse to just one
single flash point? It is as though at the moment of collapse an
instantaneous signal were sent out from the point of collapse to all
other possible collapse positions telling them not to flash. Thus
Einstein maintains (Bacciagaluppi and Valentini 2009,
p. 488),

the interpretation, according to which
|ψ|² expresses the probability that this particle is
found at a given point, assumes an entirely peculiar mechanism of
action at a distance, which prevents the wave continuously distributed
in space from producing an action in two places on the
screen.

One could see this as a tension between
local action and the description afforded by the wave function, since
the wave function alone does not specify a unique position on the
screen for detecting the particle. Einstein continues,

In
my opinion, one can remove this objection only in the following way,
that one does not describe the process solely by the Schrödinger
wave, but that at the same time one localizes the particle during
propagation.

Einstein points to Louis de Broglie's pilot wave investigations as a
possible direction to pursue if one is looking for an account of
individual processes that avoids a “contradiction with the
postulate of relativity.” He also raises the possibility not to
regard the quantum theory as describing individuals and their
processes at all and, instead, to regard it as describing only
ensembles of individuals. Indeed Einstein suggests difficulties for
any version, like de Broglie's and like quantum theory itself, that
requires representations in multi-dimensional configuration space,
difficulties that might move one further toward regarding quantum
theory as not aspiring to a description of individual systems but as
more amenable to an ensemble (or collective) point of view. Perhaps
the most important feature of Einstein's reflections at Solvay 1927 is
his insight that the clash between completeness and locality already
arises in measurements of a single variable (there, position) and does
not require measurements for an incompatible pair, as in EPR.

Following the publication of EPR Einstein set about almost
immediately to provide clear and focused versions of the argument. He
began that process within few weeks of EPR, in the June 19 letter to
Schrödinger, and continued it in an article published the
following year (Einstein 1936). He returned to this particular form of
an incompleteness argument in two later publications (Einstein 1948
and Schilpp 1949). Although these expositions differ in details they
all employ composite systems as a way of implementing indirect
measurements-at-a-distance. None of Einstein's accounts contains the
Criterion of Reality nor the tortured EPR argument
over when values of a quantity can be regarded as “elements of
reality”. The Criterion and these “elements” simply
drop out. Nor does Einstein engage in calculations, like those of
Podolsky, to fix the total wave function for the composite system
explicitly. Unlike EPR, none of Einstein's arguments makes use of
simultaneous values for complementary quantities like position and
momentum. He does not challenge the uncertainty relations. Indeed with
respect to assigning eigenstates for a complementary pair he tells
Schrödinger “ist mir wurst”—literally, it's
sausage to me; i.e., he couldn't care less. (Fine 1996, p. 38). These
writings probe an incompatibility between affirming locality and
separability, on the one hand, and completeness in the description of
individual systems by means of state functions, on the other. His
argument is that we can have at most one of these but never both. He
frequently refers to this dilemma as a “paradox”.

In the letter to Schrödinger of June 19, Einstein points to a
simple argument for the dilemma which, like the argument from the 1927
Solvay Conference, involves only the measurement of a single
variable. Consider an interaction between the Albert and Niels systems
that conserves their relative positions. (We need not worry about
momentum, or any other quantity.) Consider the evolved wave function
for the total (Albert+Niels) system when the two systems are far
apart. Now assume a principle of locality-separability (Einstein calls
it a Trennungsprinzip—separation principle): Whether a
determinate physical situation holds for Niels' system does not depend
on what measurements (if any) are made locally on Albert's system. If
we measure the position of Albert's system, the conservation of
relative position implies that we can immediately infer the position
of Niels'; i.e., we can infer that Niels' system has a determinate
position. By locality-separability it follows that Niels' system must
already have had a determinate position just before Albert began that
measurement. At that time, however, Niels' system alone does not have
a state function. There is only a state function for the combined
system and that total state function does not single out the position
we would find for Niels' system (i.e., it is not a product one of
whose factors is an eigenstate for the position of Niels'
system). Thus the description of Niels' system afforded by the quantum
state function is incomplete. A complete description would say
(definitely yes) if a determinate physical situation were true of
Niels' system. (Notice that this argument does not even depend on the
reduction of the total state function for the combined system.) In
this formulation of the argument it is clear that
locality-separability conflicts with
the eigenvalue-eigenstate link,
which holds that a quantity of a system has an eigenvalue if and only
if the state of the system is an eigenstate of that quantity with that
eigenvalue (or a mixture of such eigenstates). The “only if” part of the
link would need to be weakened in
order to interpret quantum state functions as complete descriptions
(see entry on
Modal Interpretations).

Although this simple argument concentrates on what Einstein saw as the
essentials, stripping away most technical details and distractions, he
frequently used another argument involving the measurement of more
than one quantity. (It is actually buried in the EPR paper, p. 779,
and a version also occurs in the June 19, 1935 letter to
Schrödinger. Harrigan and Spekkens, 2010 suggest reasons for
preferring a many-measurements argument.) This second argument focuses
clearly on the interpretation of quantum state functions in terms of
“real states” of a system, and not on any issues about
simultaneous values (real or not) for complementary quantities. It
goes like this.

Suppose, as in EPR, that the interaction between the two systems
preserves both relative position and zero total momentum and that the
systems are far apart. As before, we can measure either the position
or momentum of Albert's system and, in either case, we can infer a
position or momentum for Niels' system. It follows from the reduction
of the total state function that, depending on whether we measure the
position or momentum of Albert's system, Niels' system will be left
(respectively) either in a position eigenstate or in a momentum
eigenstate. Suppose too that separability holds, so that Niels' system
has some real physical state of affairs. If locality holds as well,
then the measurement of Albert's system does not disturb the assumed
“reality” for Niels' system. However, that reality appears
to be represented by quite different state functions, depending on
which measurement of Albert's system one chooses to carry out. If we
understand a “complete description” to rule out that one
and the same physical state can be described by state functions with
distinct physical implications, then we can conclude that the quantum
mechanical description is incomplete. Here again we confront a dilemma
between separability-locality and completeness. Many years later
Einstein put it this way (Schilpp 1949, p. 682);

[T]he paradox forces us to relinquish one of the following
two assertions:
(1) the description by means of the psi-function is complete
(2) the real states of spatially separate objects are independent of
each other

It appears that the central point of EPR was to argue that in
interpreting the quantum state functions we are faced with these
alternatives.

As we have seen, in framing his own EPR-like arguments for the
incompleteness of quantum theory, Einstein makes use of
separability
and
locality, which are also tacitly assumed in the EPR
paper. Using the language of “independent existence“ he
presents these ideas clearly in an article that he sent to Max Born
(Einstein 1948).

It is … characteristic of … physical objects
that they are thought of as arranged in a space-time continuum. An
essential aspect of this arrangement … is that they lay claim,
at a certain time, to an existence independent of one another, provided
these objects “are situated in different parts of space”. … The
following idea characterizes the relative independence of objects (A
and B) far apart in space: external influence on A has no direct
influence on B. (Born, 1971, pp. 170–71)

In the course of his correspondence with Schrödinger, however,
Einstein realized that assumptions about separability and locality
were not necessary in order to get the incompleteness conclusion that
he was after; i.e., to show that state functions may not provide a
complete description of the real state of affairs with respect to a
system. Separability supposes that there is a real state of affairs
and locality supposes that one cannot influence it immediately by
acting at a distance. What Einstein realized was that separability was
already part of the ordinary conception of a macroscopic object. This
suggested to him that if one looks at the local interaction of a
macro-system with a micro-system one could avoid having to assume
either separability or locality in order to conclude that the quantum
description of the whole was incomplete with respect to its
macroscopic part.

This line of thought evolves and dominates Einstein's last published
reflections on incompleteness, where he focuses on problems with the
stability of macro-descriptions rather than problems with composite
systems and locality.

the objective describability of individual macro-systems (description
of the real-state) can not be renounced without the
physical picture of the world, so to speak, decomposing into a
fog. (Einstein 1953b, p. 40. See also Einstein 1953a.)

In the August 8, 1935 letter to Schrödinger
Einstein says that he will illustrate the problem by means of a “crude macroscopic
example”.

The system is a substance in chemically unstable equilibrium, perhaps
a charge of gunpowder that, by means of intrinsic forces, can
spontaneously combust, and where the average life span of the whole
setup is a year. In principle this can quite easily be represented
quantum-mechanically. In the beginning the psi-function characterizes
a reasonably well-defined macroscopic state. But, according to your
equation [i.e., the Schrödinger equation], after the course of a
year this is no longer the case. Rather, the psi-function then
describes a sort of blend of not-yet and already-exploded
systems. Through no art of interpretation can this psi-function be
turned into an adequate description of a real state of affairs; in
reality there is no intermediary between exploded and
not-exploded. (Fine 1996, p. 78)

The point is that after a year either the gunpowder will have
exploded, or not. (This is the “real state” which in the EPR
situation requires one to assume separability.) The state function,
however, will have evolved into a complex superposition over these two
alternatives. Provided we maintain the eigenvalue-eigenstate link, the
quantum description by means of that state function will yield neither
conclusion, and hence the quantum description is incomplete. For a
contemporary response to this line of
argument, one might
look to the program of decoherence. (See
Decoherence.) That program points
to interactions with the environment which quickly reduce the
likelihood of any interference between the “exploded” and
the “not-exploded” branches of the evolved
psi-function. Then, breaking the eigenvalue-eigenstate link, one might
interpret the psi-function so that its (almost) non-interfering
branches yield a perspective according to which the gunpowder is
indeed either exploded or not. Such decoherence-based interpretations
of the psi-function are certainly “artful”, and their
adequacy is still under debate(see Schlosshauer 2007, especially
Chapter 8).

The reader may recognize the similarity between Einstein's exploding gunpowder example
and Schrödinger's cat (Schrödinger 1935a, p. 812). In the
case of the cat an unstable atom is hooked up to a lethal device that,
after an hour, is as likely to poison (and kill) the cat as not,
depending on whether the atom decays. After an hour the cat is either
alive or dead, but the quantum state of the whole atom-poison-cat
system at this time is a superposition involving the two possibilities
and, just as in the case of the gunpowder, is not a complete
description of the situation (life or death) of the cat. The
similarity between the gunpowder and the cat is hardly accidental
since Schrödinger first produced the cat example in his reply of
September 19, 1935 to Einstein's August 8 gunpowder letter. There
Schrödinger says that he has himself constructed “an example very
similar to your exploding powder keg”, and proceeds to outline the cat
(Fine 1996, pp. 82–83). Although the “cat paradox” is usually cited in
connection with the problem of quantum measurement
(see the relevant section of the entry on
Philosophical Issues in Quantum Theory)
and
treated as a paradox separate from EPR, its origin is here as an
argument for incompleteness that avoids the twin assumptions of
separability and locality. Schrödinger's development of
“entanglement”, the term he introduced as a general
description of the correlations that result when quantum systems
interact, also began in this correspondence over EPR (Schrödinger
1935a, 1935b; see
Quantum Entanglement and Information).

The literature surrounding EPR contains yet another version of the
argument, a popular version that—unlike any of
Einstein's—features the
Criterion of Reality.
Assume
again an interaction between our two systems that preserves both
relative position and zero total momentum and suppose that the systems
are far apart. If we measure the position of Albert's system, we can
infer that Niels' system has a corresponding position. We can also
predict it with certainty, given the result of the position measurement
of Albert's system. Hence, according to the Criterion of Reality, the
position of Niels' system constitutes an element of reality. Similarly,
if we measure the momentum of Albert's system, we can conclude that the
momentum of Niels' system is an element of reality. The argument now
concludes that since we can choose freely to measure either position or
momentum, it “follows” that both must be elements of reality
simultaneously.

Of course no such conclusion follows from our freedom of choice. It
is not sufficient to be able to choose at will which quantity to
measure; for the conclusion to follow from the Criterion alone one
would need to be able to measure both quantities at once. This is
precisely the point that Einstein recognized in his
1932 letter to Ehrenfest and that EPR addresses
by assuming locality and separability. What is striking about this
version is that these principles, central to the original EPR argument
and to the dilemma at the heart of Einstein's versions, are obscured
here. Instead this version features the Criterion and those
“elements of reality”. Perhaps the difficulties presented
by Podolsky's text contribute to this reading. In any case, in the
physics literature this version is commonly taken to represent EPR and
usually attributed to Einstein. This reading certainly has a prominent
source in terms of which one can understand its popularity among
physicists; it is Niels Bohr himself.

By the time of the EPR paper many of the early interpretive battles
over the quantum theory had been settled, at least to the satisfaction
of working physicists. Bohr had emerged as the
“philosopher” of the new theory and the community of
quantum theorists, busy with the development and extension of the
theory, were content to follow Bohr's leadership when it came to
explaining and defending its conceptual underpinnings (Beller 1999,
Chapter 13). Thus in 1935 the burden fell to Bohr to explain what was
wrong with the EPR “paradox”. The major article that he
wrote in discharging this burden (Bohr 1935a) became the canon for how
to respond to EPR. Unfortunately, Bohr's summary of EPR in that
article, which is the version just above, also became the canon for
what EPR contained by way of argument.

Bohr's response to EPR begins, as do many of his treatments of the
conceptual issues raised by the quantum theory, with a discussion of
limitations on the simultaneous determination of position and momentum.
As usual, these are drawn from an analysis of the possibilities of
measurement if one uses an apparatus consisting of a diaphragm
connected to a rigid frame. Bohr emphasizes that the question is to
what extent we can trace the interaction between the particle being
measured and the measuring instrument. (See Beller 1999, Chapter 7 for
a detailed analysis and discussion of the “two voices” contained in
Bohr's account.) Following the summary of EPR, Bohr (1935a, p. 700)
then focuses on the Criterion of Reality which, he says, “contains an
ambiguity as regards the meaning of the expression ‘without in any way
disturbing a system’.” Bohr agrees that the indirect measurement of
Niels' system achieved when one makes a measurement of Albert's system
does not involve any “mechanical disturbance” of Niels'
system. (Thus Bohr takes for granted that one may raise the question
of a disturbance between the two systems, and hence he takes
separability, that there are distinct systems, for granted.) Still,
Bohr claims that a measurement on Albert's system does involve
“an influence on the very conditions which define the possible
types of predictions regarding the future behavior of [Niels']
system.” What Bohr may have had in mind is that when, for
example, we measure the position of Albert's system and get a result
we can predict the position of Niels' system with certainty. However,
measuring the position of Albert's system does not allow a similarly
certain prediction for the momentum of Niels' system. The opposite
would be true had we measured the momentum of Albert's system. Thus
depending on which variable we measure on Albert's system, we will be
entitled to different sorts of predictions about the results of
further measurements on Niels' system.

There are two important things to notice about this response. The
first is this. In conceding that Einstein's indirect method for
determining, say, the position of Niels' system does not mechanically
disturb that system, Bohr departs from his original program of
complementarity, which was to base the uncertainty relations and the
statistical character of quantum theory on uncontrollable physical
interactions, interactions that were supposed to arise
inevitably between a measuring instrument and the system being
measured. Instead Bohr now
distinguishes between a genuine physical interaction (his “mechanical
disturbance”) and some other sort of “influence” on
the conditions for specifying (or “defining”) sorts of predictions for
the future behavior of a
system. In emphasizing that only the latter arise in the EPR
situation, Bohr retreats from his earlier, physically grounded
conception of complementarity.

The second important thing to notice is how Bohr's response needs to
be implemented in order to block the arguments of Einstein that pose a
dilemma between principles of locality and completeness. In Einstein's
arguments the locality principle makes explicit reference to the
reality of the unmeasured system (no immediate influence on the
reality there due to measurements made elsewhere). Hence Bohr's
pointing to an influence on conditions for specifying predictions
would not affect the argument at all unless one includes those
conditions as part of the reality of Niels' system. That would be
implausible on two counts. Firstly, it would make what is real about
Niels' system encompass what is happening to Albert's system, which is
someplace else. (Recall EPR's warning against just this move.)
Secondly, there is an issue of intelligibility. Bohr maintains that
the “conditions” (which define the possible types of
predictions regarding the future behavior of Niels' system)
“constitute an inherent element of the description of any
phenomena to which the term ‘physical reality’ can be
properly attached” (Bohr 1935a, p. 700). Thus Bohr makes the
problematic suggestion that the very expression “Niels'
system” refers to conditions for predicting the future behavior
of Niels' system. The self-reference here of “Niels'
system” generates a regress that stands in the way of
determining the conditions in question. If it were possible to bypass
the regress, then including such conditions as part of the
“reality” of the unmeasured system would automatically
preclude locality (while allowing for separability). Bohr would have
it that both systems exist (separability) but, somehow, their
existence is not independent of one another (nonlocality). If such a
conception makes sense then, by tailoring the concept of physical
reality so as make it true by definition that the quantum
theory is not local, Bohr's response might embrace separability and
even concede the validity of the EPR argument, but still block the
impact of EPR on the issue of completeness.

Despite Bohr's seeming tolerance for a breakdown of locality in his
response here to EPR, in other places Bohr rejects nonlocality in the
strongest terms. For example in discussing an electron double slit
experiment, which is Bohr's favorite model for illustrating the novel
conceptual features of quantum theory, and writing only weeks before
EPR, Bohr argues as follows.

If we only imagine the possibility that without disturbing
the phenomena we determine through which hole the electron passes, we
would truly find ourselves in irrational territory, for this would put
us in a situation in which an electron, which might be said to pass
through this hole, would be affected by the circumstance of whether
this [other] hole was open or closed; but … it is completely
incomprehensible that in its later course [the electron] should let
itself be influenced by this hole down there being open or shut. (Bohr
1935b)

It is uncanny how closely Bohr's language mirrors that of EPR. But
here Bohr defends locality and regards the very contemplation of
nonlocality as “irrational” and “completely
incomprehensible”. Since “the circumstance of whether this
[other] hole was open or closed” does affect the possible types
of predictions regarding the electron's future behavior, if we expand
the concept of the electron's “reality”, as he appears to
suggest for EPR, by including such information, we do
“disturb” the electron around one hole by opening or
closing the other hole. That is, if we give to “disturb”
and to “reality” the very same sense that Bohr appears to
give them when responding to EPR, then we are led to an
“incomprehensible” nonlocality, and into the territory of
the irrational.

There is another way of trying to understand Bohr's position.
According to one common reading (see
Copenhagen Interpretation), after EPR
Bohr embraced a relational (or contextual) account of property
attribution. On this account to speak of the position, say, of a
system presupposes that one already has put in place an appropriate
interaction involving an apparatus for measuring position (or at least
an appropriate frame of reference for the measurement; Dickson
2004). Thus “the position” of the system refers to a
relation between the system and the measuring device (or measurement
frame). In the EPR context this would seem to imply that before one is
set up to measure the position of Albert's system, talk of the
position of Niels' system is out of place; whereas after one
measures the position of Albert's system, talk of the position of
Niels' system is appropriate and, indeed, we can then say truly that
Niels' system “has” a position. Similar considerations
govern momentum measurements. It follows, then, that local
manipulations carried out on Albert's system, in a place we may assume
to be far removed from Niels' system, can directly affect what
is meaningful to say about, as well as factually true of, Niels'
system. Similarly, in the double slit arrangement, it would follow
that what can be said meaningfully and said truly about the position
of the electron around the top hole would depend on the context of
whether the bottom hole is open or shut. One might suggest that such
relational actions-at-a-distance are harmless ones, perhaps merely
“semantic”; like becoming the “best” at a task
when your only competitor—who might be miles
away—fails. Note, however, that in the case of ordinary
relational predicates it is not inappropriate (or
“meaningless”) to talk about the situation in the absence
of complete information about the relata. So you might be the best at
a task even if your competitor has not yet tried it, and you are
definitely not an aunt (or uncle) until one of your siblings gives
birth. But should we say that an electron is nowhere at all until we
are set up to measure its position, or would it be inappropriate
(meaningless?) even to ask?

If quantum predicates are
relational, they are different from many ordinary
relations in that the
conditions for the relata are taken as criterial for the application of
the term. In this regard one might contrast the relativity of
simultaneity with the proposed relativity of position. In relativistic
physics specifying a world-line fixes a frame of reference for
attributions of simultaneity to events regardless of whether any
temporal measurements are being made or contemplated. But in the
quantum case, on this proposal, specifying a frame of reference for
position (say, the laboratory frame) does not entitle one to attribute
position to a system, unless that frame is associated with actually
preparing or completing a measurement of position for that system. To
be sure, analyzing predicates in terms of occurrent measurement or
observation
is familiar from neopositivist approaches to the language of
science; for example, in Percy Bridgman's
operational analysis of physical terms, where the actual applications of
test-response pairs
constitute criteria for any meaningful use of a term
(see theory and observation in science ).
Rudolph Carnap's later introduction of reduction sentences
(see the entry on the Vienna Circle)
has a similar character. Still, this positivist reading entails
just the sort of nonlocality that Bohr seemed to abhor.

In the light of all this it is difficult to know whether a coherent
response can be attributed to Bohr reliably that would derail EPR. (In
different ways, Dickson 2004 and Halvorson and Clifton 2004 make an
attempt on Bohr's behalf. These are examined in Whitaker
2004 and Fine 2007.) Bohr may well have been aware of the difficulty in framing the
appropriate concepts clearly when, a few years after EPR, he
wrote,

The unaccustomed features of the
situation with which we are confronted in quantum theory necessitate
the greatest caution as regard all questions of terminology. Speaking,
as it is often done of disturbing a phenomenon by observation, or even
of creating physical attributes to objects by measuring processes is
liable to be confusing, since all such sentences imply a departure from
conventions of basic language which even though it can be practical for
the sake of brevity, can never be unambiguous. (Bohr 1939, p. 320.
Quoted in Section 3.2 of the entry on the Uncertainty
Principle.)

For about fifteen years following its publication, the EPR paradox
was discussed at the level of a thought experiment whenever the
conceptual difficulties of quantum theory became an issue. In 1951
David Bohm, a protégé of Robert Oppenheimer and then an
untenured Assistant Professor at Princeton
University, published
a textbook on the quantum theory in which he took a close look at EPR
in order to develop a response in the spirit of Bohr. Bohm showed how
one could mirror the conceptual situation in the EPR thought experiment
by looking at the dissociation of a diatomic molecule whose total spin
angular momentum is (and remains) zero; for instance, the dissociation
of an excited hydrogen molecule into a pair of hydrogen atoms by means
of a process that does not change an initially zero total angular
momentum (Bohm 1951, Sections 22.15–22.18). In the Bohm experiment the
atomic fragments separate after interaction, flying off in different
directions freely. Subsequently, measurements are made of their spin
components (which here take the place of position and momentum), whose
measured values would be anti-correlated after dissociation. In the
so-called singlet state of the atomic pair, the state after
dissociation, if one atom's spin is found to be positive with respect
to the orientation of an axis at right angles to its flight path, the
other atom would be found to have a negative spin with respect to an
axis with the same orientation. Like the operators for position and
momentum, spin operators for different orientations do not commute.
Moreover, in the experiment outlined by Bohm, the atomic fragments can
move far apart from one another and so become appropriate objects for
assumptions that restrict the effects of purely local actions. Thus
Bohm's experiment mirrors the entangled correlations in EPR for
spatially separated systems, allowing for similar arguments and
conclusions involving locality, separability, and completeness. Indeed, a
recently discovered note of Einstein's, that may have been prompted by
Bohm's
treatment, contains a very sketchy spin version of the EPR argument
– once again pitting completeness against locality (“A coupling
of distant things is excluded.” Sauer 2007, p. 882). Following Bohm (1951) a
paper by
Bohm and Aharonov (1957) went on to outline the machinery for a plausible
experiment in which
entangled spin correlations could be verified. It has become
customary to refer
to experimental arrangements involving determinations of spin
components for spatially separated systems, and to a variety of similar
set-ups (especially ones for measuring photon polarization), as “EPRB”
experiments—“B” for Bohm. Because of technical difficulties in
creating and monitoring the atomic fragments, however, there seem to
have been no immediate attempts to perform a Bohm version of EPR.

That was to remain the situation for almost another fifteen years,
until John Bell utilized the EPRB set-up to construct a stunning
argument, at least as challenging as EPR, but to a different conclusion
(Bell 1964). Bell shows that, under a given set of assumptions, certain
of the correlations that can be measured in runs of an EPRB experiment
satisfy a particular set of constraints, known as the Bell
inequalities. In these EPRB experiments, however, quantum theory
predicts that the measured correlations violate the Bell inequalities,
and by an experimentally significant amount. Thus Bell shows (see the
entry on Bell's Theorem)
that quantum theory is inconsistent with the
given assumptions. Prominent among these is an assumption of locality,
similar to the locality assumptions tacitly assumed in EPR and
(explicitly) in the one-measurement and many-measurement arguments of
Einstein that depend on separability-locality. Thus Bell's theorem is
often characterized as showing that quantum theory is
nonlocal. However, since several other assumptions are needed in any
derivation of the Bell inequalities (roughly, assumptions guaranteeing
a classical representation of the quantum probabilities; see Fine
1982a, and Malley 2004), one should be cautious about singling out
locality as necessarily in conflict with the quantum theory.

Bell's results were explored and deepened by various theoretical
investigations and they have stimulated a number of increasingly
sophisticated and delicate EPRB-type experiments designed to test
whether the Bell inequalities hold where quantum theory predicts they
should fail. With a few anomalous exceptions, the experiments confirm
the quantum violations of the inequalities. (Baggott 2004 contains a
readable account of the major refinements and experiments. Genovese
2005 is an exhaustive technical review.) The confirmation is
quantitatively impressive, although the experiments continue to leave
open at least two different ways (corresponding to the prism and
synchronization models sketched in Fine 1982b) to reconcile the data
with frameworks that embody locality and separability. One way
(prisms) exploits the low rate of detection in most experiments; the
other way (synchronization) exploits time delays associated with
coincidence counts. (See Larsson 1999, and Szabo and Fine 2002 for the
former and for the latter Larsson and Gill 2004 and the EPRB
simulation constructed in de Raedt et al 2007.) The
difficulty is to carry out an efficient experiment that controls for
these sorts of errors and that excludes communication about detections
between the two wings of the experiment as well as communication
between emissions at the source and the choice of measurements in the
wings. (Scheidl et al 2010 is an attempt to exclude these two
types of communication but does not control the errors sufficiently,
and Giustina et al 2013 is an attempt to control the errors
but leaves open the possibility of communication.) While the exact
significance of experimental tests of the Bell inequalities thus
remains somewhat controversial, the techniques developed in the
experiments, and related theoretical ideas for utilizing the
entanglement associated with EPRB-type interactions, have become
important in their own right. These techniques and ideas, stemming
from EPRB and the Bell theorem, have applications now being advanced
in the relatively new field of quantum information theory —
which includes quantum cryptography, teleportation and computing (see
Quantum Entanglement and Information).

To go back to the EPR dilemma between
locality
and completeness, it would appear from the Bell theorem that Einstein's
strategy of maintaining locality, and thereby concluding that the
quantum description is incomplete, may have fixed on the wrong horn.
Even though the Bell theorem does not rule out locality conclusively,
it should certainly make one wary of assuming it. On the other hand,
since Einstein's
exploding gunpowder argument
(or Schrödinger's cat), along with his later arguments over
macro-systems, support incompleteness without assuming locality, one
should be wary of adopting the other horn of the dilemma, affirming
that the quantum state descriptions are complete and
“therefore” that the theory is nonlocal. It may well turn
out that both horns need to be rejected: that the state functions do
not provide a complete description and that the theory is also
nonlocal (although possibly still separable; see Winsberg and Fine
2003). There is at least one well-known approach to the quantum theory
that makes a choice of this sort, the de Broglie-Bohm approach
(Bohmian Mechanics). Of course it may also
be possible to break the EPR argument for the dilemma plausibly by
questioning some of its other assumptions (e.g.,
separability,
the reduction postulate, the
eigenvalue-eigenstate link, or a common assumption
of measurement independence). That might free up the remaining
option, to regard the theory as both local and complete. Perhaps a
well-developed version of the
Everett Interpretation would come to occupy
this branch of the interpretive tree.